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Evaluate the following : limx→∞[ax3+bx2+cx+dex3+fx2+gx+h] - Mathematics and Statistics

Sum

Evaluate the following :

`lim_(x -> ∞) [("a"x^3 + "b"x^2 + "c"x + "d")/("e"x^3 + "f"x^2 + "g"x + "h")]`

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Solution

`lim_(x -> ∞) [("a"x^3 + "b"x^2 + "c"x + "d")/("e"x^3 + "f"x^2 + "g"x + "h")]`

= `lim_(x -> ∞)[(("a"x^3 + "b"x^2 + "c"x + "d")/(x^3))/(("e"x^3 + "f"x^2 + "g"x + "h")/(x^3))]  ...[("Divide numerator and"),("denominator by"  x^3)]`

= `(lim_(x -> ∞)("a" + "b"/x + "c"/x^2 + "d"/x^3))/(lim_(x -> ∞)("e" + "f"/x + "g"/x^2 + "h"/x^3)`

= `(lim_(x -> ∞) "a" + lim_(x -> ∞) "b"/x + lim_(x -> ∞) "c"/x^2 + lim_(x -> ∞) "d"/x^3)/(lim_(x-> ∞)"e" + lim_(x -> ∞) "f"/x + lim_(x -> ∞) "g"/x^2 + lim_(x -> ∞) "h"/x^3)`

= `("a" + 0 + 0 + 0)/("e" + 0 + 0 + 0)  ...[lim_(x -> ∞) 1/x^"k" = 0, "k" > 0]`

= `"a"/"e"`

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